C₂×#1083840 #72294549

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#1083840

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(4,6)(7,8)(9,13)(10,12)(11,14)(15,16,18,21)(17,20,22,19)(24,28)(25,30);(7,8,10,12)(9,11,14,13)(15,16,18,21)(17,20,22,19)(23,24)(25,26)(27,28)(29,30);(3,4)(5,6)(15,17)(16,19)(18,22)(20,21)(23,25)(24,29)(26,28)(27,30);(7,9)(8,11)(10,14)(12,13)(15,17)(16,20)(18,22)(19,21)(23,25)(24,26)(27,30)(28,29);(7,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,21)(20,22)(23,26)(24,30)(25,28)(27,29);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937
maximal normal subgroup of	C₂×#72293937
maximal quotient of	C₂×#72293986

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,6)(7,8)(9,13)(10,12)(11,14)(15,16,18,21)(17,20,22,19)(24,28)(25,30), (7,8,10,12)(9,11,14,13)(15,16,18,21)(17,20,22,19)(23,24)(25,26)(27,28)(29,30), (3,4)(5,6)(15,17)(16,19)(18,22)(20,21)(23,25)(24,29)(26,28)(27,30), (7,9)(8,11)(10,14)(12,13)(15,17)(16,20)(18,22)(19,21)(23,25)(24,26)(27,30)(28,29), (7,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,21)(20,22)(23,26)(24,30)(25,28)(27,29), (1,2));

C2×#1083840 #72294549